ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
EXPONENTIAL ESTIMATES FOR QUANTUM GRAPHS
SETENAY AKDUMAN, ALEXANDER PANKOV Communicated by Vicentiu D. Radulescu
Abstract. The article studies the exponential localization of eigenfunctions associated with isolated eigenvalues of Schr¨odinger operators on infinite metric graphs. We strengthen the result obtained in [3] providing a bound for the rate of exponential localization in terms of the distance between the eigenvalue and the essential spectrum. In particular, if the spectrum is purely discrete, then the eigenfunctions decay super-exponentially.
1. Introduction
A quantum graph is a metric graph equipped with a self-adjoint Hamiltonian.
For a comprehensive introduction to quantum graphs we refer to [4, 6, 11, 12] and references therein). Other aspects of differential equations on graphs and networks are available in [13, 14, 15] and references therein.
Typically, Hamiltonians are operators of Schr¨odinger type generated by the second-order differential expression
−d2
dx2 +V(x)
on the edges of graph and certain conditions at the vertices. In this paper we use the Kirchhoff vertex conditions and impose sufficiently weak assumptions on the potentialV under which the operator is self-adjoint and bounded below.
Our main concern in this paper is the exponential localization of eigenfunctions associated with isolated eigenvalues. In the case of classical Schr¨odinger opera- tors this topic goes back to Schnol’s paper [20] (see also [7]). For one-dimensional operators similar results were obtained in [16, 21]. The current state of the art of the topic is reviewed in [10, 18]. The first localization result for operators on metric graphs is obtained in [3]. The approach in that paper relies upon an elemen- tary perturbation theory for linear operators and provides the exponential decay of eigenfunctions with sufficiently small rate. Papers [8, 9] are devoted to an extension of Agmon’s geometric approach to quantum graphs.
In this article we obtain a stronger result on exponential decay of eigenfunction than in [3]. We provide a bound for the rate of decay in terms of the distance between the associated eigenvalue and the essential spectrum. Though not optimal,
2010Mathematics Subject Classification. 34B45, 34L40, 81Q35.
Key words and phrases. Infinite metric graph; Schr¨odinger operator; eigenfunction;
exponential decay.
c
2018 Texas State University.
Submitted July 14, 2018. Published September 10, 2018.
1
the bound is strong enough to imply that all eigenfunctions decay superexponentiall fast provided that the spectrum is purely discrete. The techniques relies upon estimates of the derivative of solution to the Schr¨odinger equation in terms of the solution itself on certain special domains. Such domains, called quasi-balls and quasi-annuli, are defined in terms of properly regularized distance function introduced in [3]. This approach can be considered as a suitable variant of the original Schnol’s method [20]. Also it permits us to obtain an extension to quantum graphs for another Schnol’s result [20] that provides an estimate for the distance from a real number to the spectrum in terms of exponential growth of solution to the Schr¨odinger equation, and known as Schnol’s theorem [18] in the classical setting.
As consequence, we have a condition for a point to belong to the spectrum in terms of such solutions. Under some stronger assumptions the last result is obtained in [12]. As an application, we give sufficient conditions under which eigenfunctions belong to allLp spaces.
This article is organized as follows. In Section 2 we recall basic information about metric graphs and Schr¨odinger operators on them. Section 3 is the technical core of the paper. In Section 4 we prove the main results while Section 5 is dealing with some consequences of the main results.
2. Metric graphs and Schr¨odinger operators
Let us consider a graph Γ = (E, V) with countably infinite sets of edges E and verticesV. We allow loops and multiple edges, and assume that the graph is connected, i.e., any two vertices are terminal vertices of a path of edges. Recall that the degree deg(v) of a vertex v ∈V is the number of edges emanating from v. We assume that all vertices of Γ have finite degrees which are positive due to the connectedness of Γ. For any vertex v ∈ V we denote by Ev the set of edges adjacent tov.
The graph Γ is said to be ametric graphif each edgeeis identified with an interval [0, le] of real line. We always assume that there exist two positive constants l and l such that
l≤le≤l (2.1)
for alle∈E. Ife∈E, we denote byxethe induced coordinate ofe(we often skip the indexein this notation). The same symbolxis often used for a point on Γ.
The distanced(x, y) between two pointsxandyin Γ is defined as the length of a shortest path that connects these points. Furthermore, there is a natural measure, dx, on Γ which coincides with the Lebesgue measure on each edge. Thus, Γ is a non-compact metric measure space. We fix an arbitrary vertex o ∈V considered as an origin and set
d(x) =d(x, o). (2.2)
We utilize the standard notationLp(Γ), 1≤p≤ ∞, for the Lebesgue spaces on Γ with respect to the measure dx. The norm in a Banach space E is denoted by k · kE, and we set k · k =k · kL2. The space Lploc(Γ), 1≤ p≤ ∞, consists of all measurable functionsf on Γ such thatf|e∈Lp(e) for all e∈E.
The Sobolev spaceH1(Γ) consists of allcontinuouscomplex valued functionsf on Γ such thatf|e∈H1(e) for all edgese∈Eand
kfk2H1 =X
e∈E
kfk2H1(e)<∞.
For every function f ∈ H1(Γ) we have f(x) → 0 as x → ∞ in the sense that d(x)→ ∞. Furthermore, there is a continuous, dense embedding H1(Γ)⊂Lp(Γ) ifp≥2.
The space BS(Γ) of Stepanov bounded functions (known also under the name uniformL1 space [18]) consists of all functionsf ∈L1loc(Γ) such that
kfkBS= sup
e∈E
kfkL1(e)<∞.
We need the following inequality (see [2, Lemma 2.1]). For everyε >0, Z
Γ
|f(x)||u(x)|2dx≤ kfkBS εku0k2+ (ε−1+l−1)kuk2
, (2.3)
wheneverf ∈BS(Γ) andu∈H1(Γ).
LetV(x) be a real function on Γ. Throughout this paper we accept the following assumption
(A1) The functionV is locally integrable on Γ andV−∈BS(Γ).
Here and thereafter we use the notationa+= max[a,0] anda−=−min[a,0].
We consider the Schr¨odinger operatorL associated with the differential expres- sion,
L=−d2
dx2+V(x)
together with certain vertex conditions. The domain D(L) of L consists of all u∈L2(Γ) such thatuand u0 are absolutely continuous on each edge of Γ (hence, u00∈L1loc(Γ)),
uis continuous at all vertices of Γ, (2.4) X
e∈Ev
du dne
(v) = 0 (2.5)
for all verticesv∈V, wherednd
e stands for the outward derivatives at the endpoints of the edgee, andLu∈L2(Γ). Then the action ofL is defined byLu=Lufor all u∈D(L). As shown in [2], Lis a densely defined, self-adjoint operator in L2(Γ).
Furthermore,Lis bounded below andD(L)⊂H1(Γ). Notice that conditions (2.4) and (2.5) are calledKirchhoff vertex conditions. Alternatively, the operatorL can be defined in terms of quadratic forms [2].
Note that the distance functiondis not smooth and does not satisfy the Kirchhoff vertex conditions. To overcome this difficulty, we need the following lemma (see [3, Lemma 4.1]).
Lemma 2.1. There exists a function η ∈C(Γ×Γ) such that for everyy ∈Γ the functionη(·, y)belongs toC2(e)on each edgee, its first and second derivatives with respect to the first variable are bounded onΓuniformly with respect toy,η satisfies the Kirchhoff vertex conditions with respect to the first variable, and
d(x, y)−c0≤η(x, y)≤d(x, y) +c0, (x, y)∈Γ×Γ, (2.6) withc0>0independent of (x, y).
To abbreviate we setη(x) =η(x, o).
3. Preliminary results
First we provide a corrected version of [7, Theorem 10, Section 3] (no proof is given there).
Lemma 3.1. LetAbe a self-adjoint operator in a Hilbert spaceH, with the domain D(A), λ0 ∈ R, and δ > 0. The spectral subspace of A that corresponds to the interval [λ0−δ, λ0+δ]is infinite dimensional if and only if there exists a sequence un∈D(A)such thatkunk= 1,un→0weakly inH andkAun−λ0unk ≤δfor all n.
Proof. Without loss, we may assume thatλ0= 0.
(a) Sufficiency. Let ∆ = [−δ, δ]. Assume that dimE(∆)H =∞. Then there exists an orthonormal sequenceun ∈D(A)∩E(∆)H such thatun →0 weakly and
kAunk2= Z
∆
λ2d(E(λ)un, un)≤ Z
∆
δ2d(E(λ)un, un) =δ2kunk2=δ2. (b) Necessity. Suppose the contrary. Thenσ(A)∩∆ consists of finite number of isolated eigenvalues of finite multiplicity. Therefore, there existsδ1> δ such that
σ(A)∩∆1=σ(A)∩∆, where ∆1= [−δ1, δ1]. Then
δ12=δ12kunk2
= Z
∆1
δ12d(E(λ)un, un) + Z
R\∆1
δ12d(E(λ)un, un)
≤ Z
∆1
δ12d(E(λ)un, un) + Z
R\∆1
λ2d(E(λ)un, un)
≤ Z
∆1
δ12d(E(λ)un, un) + Z
R
λ2d(E(λ)un, un)
=δ12kE(∆1)unk2+kAunk2
=δ12kE(∆)unk2+kAunk2
≤δ12kE(∆)unk2+δ2.
Since un → 0 weakly and E(∆)H is finite dimensional, then kE(∆)unk → 0.
Passing to the limit, we obtain thatδ1≤δ, a contradiction.
Remark 3.2. We recall an easy consequence of the spectral theorem. If A is a self-adjoint operator andσ(A)∩[λ0−δ, λ0+δ] =∅, then
kAu−λ0uk> δkuk for allu∈D(A),u6= 0.
Givenx0∈Γ andR >0, we introduce balls
B(x0, R) ={x∈Γ :d(x, x0)≤R}
and quasi-balls
Ω(x0, R) ={x∈Γ :η(x, x0)≤R}.
If x0 = 0, we use the abbreviations B(R) and Ω(R), respectively. For R > c0, inequality (2.6) implies that
B(x0, R−c0)⊂Ω(x0, R)⊂B(x0, R+c0). (3.1)
A solution of equation
−u00+V(x)u−λu= 0, (3.2)
is a functionuon Γ such thatuand u0 are absolutely continuous on each edge of Γ and (3.2) holds almost everywhere.
Lemma 3.3. Let R1< R andx0∈Γ. Ifuis a solution of (3.2)that satisfies the Kirchhoff vertex conditions, then
Z
Ω(x0,R1)
|u0(x)|2dx≤C(k(V −λ)−kBS+ 1)2 Z
Ω(x0,R)
|u(x)|2dx, (3.3) whereC >0 depends onR−R1 but not onx0 andλ.
Proof. Without loss of generality, we assume that λ = 0. Letψ(r), r ∈ R, be a smooth function such that 0≤ψ(r)≤1 for allr∈R,ψ(r) = 1 forr≤R1,ψ(r) = 0 forr≥R, and|ψ0(r)|and|ψ00(r)|are bounded by a constant that depends only on R−R1. By Lemma 2.1, the function
ϕ(x) =ψ(η(x, x0))
is smooth on every edge of Γ and satisfies Kirchhoff vertex conditions (2.4) and (2.5).
Sinceu(x) andϕ2(x)u(x) satisfy the Kirchhoff condition, and suppϕ2u⊂Ω(x0, R), integration by parts implies
0 = Z
Γ
(Lu)(ϕ2u)dx= Z
Γ
u0(ϕ2u)0+V(x)(ϕu)2 dx
= Z
Γ
(ϕ2)0uu0+ϕ2(u0)2+V(x)(ϕu)2 dx
(3.4)
and Z
Γ
(ϕ2)0uu0dx= 1 2 Z
Γ
(ϕ2)0(u2)0dx=−1 2
Z
Γ
(ϕ2)00u2dx . (3.5) It follows from (3.4) and (3.5) that
Z
Γ
ϕ2(u0)2dx= 1 2
Z
Γ
(ϕ2)00u2dx− Z
Γ
V(x)(ϕu)2dx . Hence,
kϕu0k2≤ 1
2k |(ϕ2)00|1/2uk2+ Z
Γ
V−(x)(ϕu)2dx . (3.6) By inequality (2.3), for any ε > 0 the integral in the right-hand side of (3.6) is bounded above by
kV−kBS(εkϕu0+ϕ0uk2+ (ε−1+l−1)kϕuk2). Takingε= 1/(4kV−kBS), using the inequality
kϕu0+ϕ0uk2≤2(kϕu0k2+kϕ0uk2), and estimatingkϕ0uk andk|(ϕ2)00|1/2ukin terms of
Z
Ω(x0,R)
|u(x)|2dx ,
from (3.6), we obtain
kϕu0k2≤C(kV−kBS+ 1)2 Z
Ω(x0,R)
|u(x)|2dx .
Since the left-hand side of (3.3) does not exceedkϕu0k2, the result follows.
Also we need an estimate of type (3.3) on quasi-annuli Ω(x0, R0, R) ={x∈Γ :R0≤η(x, x0)≤R}.
Lemma 3.4. Let R0 < R01< R1< Rand x0 ∈Γ. Ifuis a solution of (3.2) that satisfies the Kirchhoff vertex conditions, then
Z
Ω(x0,R01,R1)
|u0(x)|2dx≤C(k(V −λ)−kBS+ 1)2 Z
Ω(x0,R0,R)
|u(x)|2dx,
whereC >0 depends onR−R1 andR01−R0 but not onx0 andλ.
Proof. We follow the same arguments as in the proof of Lemma 3.3. The main difference is that now we choose a smooth function ψ(r), r ∈ R, such that 0 ≤ ψ(r) ≤1 for all r ∈ R, ψ(r) = 1 if R01 ≤r ≤ R1 and ψ(r) = 0 if either r ≤R0 or r ≥R. The function ψ can be chosen in such a way that its first and second derivatives are bounded by a constant that depends only onR−R1and R01−R0. Then we use the test functionϕ2(x)u(x), whereϕ(x) =ψ(η(x)).
4. Exponential estimates We begin with the exponential decay of eigenfunctions.
Theorem 4.1. There exists a constantc >0, independent ofV, with the following property. If u∈L2(Γ)is an eigenfunction of L associated with an isolated eigen- value λof finite multiplicity, andκis the distance from λtoσess(L), then for any α >0such that
α <ln
1 + κ2
c(k(V −λ)−kBS+ 1)2
then
|u(x)| ≤Cαe−α2d(x), x∈Γ, (4.1) for some Cα >0. If, in addition, σ(L)is purely discrete, then (4.1) holds for all α >0.
Proof. Without loss of generality, we assume thatλ= 0. Consider the function J(r) =
Z
Ωc(r)
|u(x)|2dx ,
where the superscriptc stands for the complement of a subset in Γ. LetA be the set of allαsuch that
J(r)≤Ce−αr, r >0, (4.2) with someC=C(α)>0, and letα0= supA.
Assume thatα0 6= +∞. Then for every δ > 0 and everyC >0 there exists a sequencern→ ∞satisfying
J(rn)> Ce−(α0+δ)rn. (4.3) As a consequence, givenδ >0, there exists a sequenceρn → ∞such that
J(ρn)≤eα0+δJ(ρn+ 1). (4.4) Indeed, if this is not so, then
J(r)< e−(α0+δ)J(r−1)
for allr > r0. Iterating this inequality, we obtain that J(r)< Ce−(α0+δ)r for someC >0 which contradicts (4.3).
Now we choose a smooth functionϕonRsuch that 0≤ϕ(r)≤1 for allr∈R and
ϕ(r) =
(0 forr≤1/4 1 forr≥3/4
and set ϕn(x) = ϕ(η(x)−ρn). Then we define the functions un(x) =ϕn(x)u(x) and vn(x) =kunk−1un(x). By Lemma 2.1, both functions un and vn satisfy the Kirchhoff vertex conditions. Notice that kvnk = 1 and suppvn ⊂ Ωc(ρn+ 1/4).
Hence,vn→0 weakly inL2(Γ). It is easily seen that
kun(x)k2≥J(ρn+ 1)≥e−(α0+δ)J(ρn). (4.5) On the other hand,
Lun=−ϕ00un−2ϕ0u0n. Therefore,
kLunk ≤C1
nZ
Ω(ρn+14,ρn+34)
|u0|2dxo1/2 +C2
nZ
Ω(ρn,ρn+1)
|u|2dxo1/2 .
By Lemma 3.4 and inequalities (4.4) and (4.5), kLunk2≤a
Z
Ω(ρn,ρn+1)
|u|2dx
=a[J(ρn)−J(ρn+1)]≤a[J(ρn)−e−(α0+δ)J(ρn)]
=aJ(ρn){1−e−(α0+δ)} ≤ae(α0+δ){1−e−(α0+δ)}kunk2
≤a{eα0+δ−1}kunk2, wherea=c(kV−kBS+ 1)2. Hence,
kLvnk ≤a(eα0+δ−1). Using Lemma 3.1, we conclude that
κ2≤a(eα0+δ−1).
Sinceδ >0 is arbitrary, it follows that κ2≤a(eα0−1) and α0≥ln
1 +κ2 a
. As consequence, for anyα <ln 1 +κa2
there exists a constantC=C(α) such that Z
Ωc(r)
|u(x)|2dx≤Ce−αr (4.6) for all r ≥0 provided that α0 < ∞. If α0 = ∞, inequality (4.6) holds trivially.
Finally, the previous argument shows that ifκ=∞, thenα0=∞, and inequality (4.6) holds in all possible cases.
Now we show that integral estimate (4.6) implies uniform decay of (4.1). The inclusion Ω(r)⊂B(r+c0) implies thatBc(r+c0)⊂Ωc(r). As consequence,
Z
Bc(r+c0)
|u(x)|2dx≤Ce−αr (4.7) for allr≥0. Since, by (3.1),
B(x0, R−c0)⊂Ω(x0, R)⊂B(x0, R+c0), we have that, for anyy∈Γ,
B(y,¯l)⊂Ω(y,¯l+c0)⊂Ω(y,¯l+c0+ 1)⊂B(y,¯l+ 2c0+ 1), where ¯l is defined by (2.1). If
d(y) =d(y, o)>(r+c0) + (¯l+ 2c0+ 1) =r+ ¯l+ 3c0+ 1, thenB(y,¯l+ 2c0+ 1)∩B(r+c0) =∅ and, hence,
Ω(y,¯l+c0+ 1)⊂B(y,¯l+ 2c0+ 1)⊂Bc(r+c0). By (4.7),
Z
Ω(y,¯l+c0+1)
|u(x)|2dx≤Ce−αr, (4.8) and, by Lemma 3.3,
Z
B(y,¯l)
|u0(x)|2dx≤ Z
Ω(y,¯l+c0)
|u0(x)|2dx
≤C1
Z
Ω(y,¯l+c0+1)
|u(x)|2dx
≤C2e−αr.
(4.9)
Since all edges have length less than or equal to ¯l, there is an edge e⊂B(y,¯l) that containsy. By (4.9),
Z
e
|u0(x)|2dx≤C2e−αr, while (4.6) yields
Z
e
|u(x)|2dx≤Ce−αr. Hence,
kuk2H1(e)≤C3e−αr.
Since the length le satisfies le ≥l > 0, then the embedding constant of H1(e)⊂ L∞(e) is independent ofle. As consequence,
|u(y)| ≤C4e−αr/2. Now, we takey∈Γ such that
ρ=d(y) =r+ ¯l+ 3c0+ 2. Thenr=ρ−λ¯−3c0−2, and
|u(y)| ≤Cee −αρ/2, where
Ce=C4eα2(¯l+3c0+2).
This completes the proof.
Corollary 4.2. Assume that the spectrum ofLis purely discrete. Ifuis an eigen- function ofL, then for anyα >0 there existsCα>0 such that
|u(x)| ≤Cαe−αd(x), x∈Γ.
Now we provide an estimate for the distance between λ∈Rand the spectrum σ(L) in terms of solutions to equation (3.2).
Theorem 4.3. There exists a constantc >0, independent ofV, with the following property. Suppose that u6= 0 is a solution of equation (3.2)on Γ that satisfies the Kirchhoff vertex conditions and
Z
B(r)
|u(x)|2dx≤Ceαr (4.10) for some α > 0 and C > 0, then the distance of the point λ from σ(L) does not exceed
c(k(V −λ)−kBS+ 1)(eα−1)1/2.
In particular, if (4.10) holds for allα >0with C=Cα>0, thenλ∈σ(L).
Proof. Without lost of generality assume thatλ= 0. By (3.1), Z
B(r−c0)
|u(x)|2dx≤ Z
Ω(r)
|u(x)|2dx≤ Z
B(r+c0)
|u(x)|2dx .
Hence, the function
J(r) = Z
Ω(r)
|u(x)|2dx
satisfiesJ(r)≤Ceαr for someC >0.
For any givenδ >0, there exists a sequenceρn → ∞such that
J(ρn+1)< eα+δJ(ρn). (4.11) If not, then
J(r)> eα+δJ(r−1)
for all sufficiently larger. Iterating this inequality, we obtain that J(r)≥Ce(α+δ)r,
withC >0, which is incompatible with (4.10).
As in the proof of Theorem 4.1, we choose a smooth functionϕonRsuch that 0≤ϕ(r)≤1 for allr∈Rand
ϕ(r) =
(0 forr≤1/4 1 forr≥3/4, and setϕn(x) =ϕ(η(x)−ρn). Then we define the function
un(x) ={1−ϕn(x)}u(x).
Note that suppun ⊂Ωc(ρn+ 1/4), andun satisfies the Kirchhoff conditions. It is easily seen that
kun(x)k2≥J(ρn). (4.12)
On the other hand,
Lun=−ϕ00nu−2ϕ0nu0
and, hence,
kLunk ≤C1nZ
Ω(ρn+14,ρn+34)
|u0|2dxo1/2
+C2nZ
Ω(ρn,ρn+1)
|u|2dxo1/2
.
Using Lemma 3.4 and inequalities (4.11) and (4.12), we obtain kLunk2≤C(kV−kBS+ 1)2
Z
Ω(ρn,ρn+1)
|u(x)|2dx,
=C(kV−kBS+ 1)2{J(ρn+1)−J(ρn)}
≤C(kV−kBS+ 1)2(eα+δ−1)J(ρn)
≤C(kV−kBS+ 1)2(eα+δ−1)kunk2. Thus,
kLunk ≤c(kV−kBS+ 1)(eα+δ−1)1/2kunk, where c=√
C. From this inequality and Remark 3.2, it follows that the distance to the pointλfromσ(L) does not exceed
c(kV−kBS+ 1)(eα+δ−1)1/2,
and since the numberδ >0 is arbitrary, the result follows.
5. Applications
In this section we make an additional assumption. Namely, we assume that there existµ >0 andCµ >0 such that for all r >0,
|B(r)| ≤Cµeµr, (5.1) where |S| is the measure of S ⊂ Γ. The infimum of all suchµ is denoted by µ0. Ifµ0 >0, the graph Γ is ofexponential growth. Otherwise, ifµ0= 0, then Γ is of sub-exponential growth.
Letube a continuous function on Γ such that
|u(x)| ≤Cαe−αd(x)
with positive constantsαandCα. Ifp∈[0,∞) andµ < αp, withµfrom inequality (5.1), then
Z
Γ
|u(x)|pdx≤
∞
X
n=1
Z
B(n)\B(n−1)
|u(x)|pdx≤C
∞
X
n=1
e−(αp−µ)n<∞,
and, hence, u ∈ Lp(Γ). Together with Theorem 4.1, this implies the following results.
Corollary 5.1. Assume that Γ is of sub-exponential growth. If u ∈ L2(Γ) is an eigenfunction associated with an isolated eigenvalue of finite multiplicity, then u∈Lp(Γ)for allp∈[1,∞].
Corollary 5.2. Assume that Γ is of exponential growth and the spectrum ofL is purely discrete. If u ∈ L2(Γ) is any eigenfunction of L, then u ∈ Lp(Γ) for all p∈[1,∞].
Note that these statements are non-trivial only in the case whenp∈[1,2) because u∈H1(Γ)⊂Lp(Γ) ifp∈[2,∞]. The following statement is an easy consequence of Theorem 4.3.
Corollary 5.3. Let u6= 0 be a solution of (3.2) on Γ that satisfies the Kirchhoff vertex conditions and, for someβ >0 andCb>0,
|u(x)| ≤Cβeβd(x),
and β0 is the infimum of all such β. Then the distance of the point λ from σ(L) does not exceed
c(k(V −λ)−kBS+ 1)(e2β0+µ0−1)1/2,
where the constant c >0 is independent ofV. In particular, if β0 =µ0 = 0, then λ∈σ(L).
In Corollary 5.3,β0=µ0= 0 means that both the graph Γ and the solutionu are of sub-exponential growth. Also we point out that ifuis bounded, thenβ0= 0.
Acknowledgements. A. Pankov was supported by Simons Foundation, Award 410289.
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Setenay Akduman
Department of Mathematics, Izmir Democracy University, Izmir, 35140, Turkey E-mail address:[email protected]
Alexander Pankov
Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA E-mail address:[email protected]
